A person observed that he required 30 seconds less time to cross a circular ground along its diameter than to cover it once along the boundary. If his speed was 30 m/minute, then the radius of the circular ground is `(" Take " pi = 22/7)` :

Options

A. 5.5 m
B. 7.5 m
C. 10.5 m
D. 3.5 m

Answer

To solve the problem, we need to determine the radius of the circular ground based on the information provided.

### Step-by-Step Solution:

1. **Define Variables**:
Let the radius of the circular ground be \( r \) meters.

2. **Calculate the Diameter**:
The diameter \( d \) of the circular ground is given by:
\[ d = 2r \]

3. **Calculate the Circumference**:
The circumference \( C \) of the circular ground is given by:
\[ C = 2\pi r \]
Substituting \( \pi = \frac{22}{7} \):
\[ C = 2 \times \frac{22}{7} \times r = \frac{44r}{7} \]

4. **Calculate Time to Cross the Diameter**:
The time \( t_d \) taken to cross the diameter at a speed of 30 m/min is:
\[ t_d = \frac{d}{\text{speed}} = \frac{2r}{30} \]

5. **Calculate Time to Cover the Boundary**:
The time \( t_c \) taken to cover the circumference at the same speed is:
\[ t_c = \frac{C}{\text{speed}} = \frac{\frac{44r}{7}}{30} = \frac{44r}{210} = \frac{22r}{105} \]

6. **Set Up the Equation**:
According to the problem, the time to cross the diameter is 30 seconds less than the time to cover the boundary:
\[ t_c - t_d = 30 \]
Substituting the expressions for \( t_c \) and \( t_d \):
\[ \frac{22r}{105} - \frac{2r}{30} = 30 \]

7. **Find a Common Denominator**:
The common denominator for 105 and 30 is 210. Rewrite the equation:
\[ \frac{22r \times 2}{210} - \frac{2r \times 7}{210} = 30 \]
This simplifies to:
\[ \frac{44r - 14r}{210} = 30 \]
Which further simplifies to:
\[ \frac{30r}{210} = 30 \]

8. **Solve for \( r \)**:
Multiply both sides by 210:
\[ 30r = 6300 \]
Divide by 30:
\[ r = 210 \]

9. **Final Answer**:
The radius of the circular ground is \( \boxed{210} \) meters.